Homogenization of nonconvex unbounded singular integrals - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Annales Mathématiques Blaise Pascal Année : 2017

Homogenization of nonconvex unbounded singular integrals

Résumé

We study periodic homogenization by $\Gamma $-convergence of integral functionals with integrands $W(x,\xi )$ having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to infinity as the volume of matter tends to zero, i.e., $W(x,\xi )=\sum _{j\in J}\mathbf{1}_{V_j}(x)H_j(\xi )$ where $\lbrace V_j\rbrace _{j\in J}$ is a finite family of open disjoint subsets of $\mathbb{R}^N$, with $|\partial V_j|=0$ for all $j\in J$ and $|\mathbb{R}^N\setminus \bigcup _{j\in J}V_j|=0$, and, for each $j\in J$, $H_j(\xi )\rightarrow \infty $ as $\det \xi \rightarrow 0$. In fact, our results apply to integrands of type $W(x,\xi )=a(x)H(\xi )$ when $H(\xi )\rightarrow \infty $ as $\det \xi \rightarrow 0$ and $a\in L^\infty (\mathbb{R}^N;[0,\infty [)$ is $1$-periodic and is either continuous almost everywhere or not continuous. When $a$ is not continuous, we obtain a density homogenization formula which is a priori different from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.
Fichier principal
Vignette du fichier
Art_Mandallena_al_Homogenization_nonconvex_integrals_2017.pdf (842.07 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte
Loading...

Dates et versions

hal-01644535 , version 1 (13-12-2017)

Identifiants

Citer

Omar Anza Hafsa, Nicolas Clozeau, Jean-Philippe Mandallena. Homogenization of nonconvex unbounded singular integrals. Annales Mathématiques Blaise Pascal, 2017, 24 (2), pp.135-193. ⟨10.5802/ambp.367⟩. ⟨hal-01644535⟩
430 Consultations
116 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More